∫dm=M()massofthebody
r
M
cm = ∫rdm
1
.
Note: If an object has symmetric uniform mass
distribution about x axis then y coordinate of COM is
zero and vice-versa
Centre of Mass of a Uniform Rod
Suppose a rod of mass M and length L is lying along the
x-axis with its one end at x = 0 and the other at x = L.
Mass per unit length of the rod = M/L
Hence, dm, (the mass of the element dx situated at
x = x is)=
M
L
dx
e coordinates of the element PQ are (x, 0, 0).
erefore, x-coordinate of COM of the rod will be
x
xdm
dm
x
M
L
dx
ML
xdx
L
L L
L
cm ==
==
∫
∫
∫
∫
0 0
0
1
2
()
e y-coordinate of COM is y
ydm
cm dm
==∫
∫
0
Similarly, zcm = 0
i.e., the coordinates of COM of the rod are
L
2
,, 00.
^
Or it lies at the centre of the rod.
RIGID BODY
A rigid body is one whose geometric shape and size
remains unchanged under the action of any external
force. When a rigid body performs rotational motion,
the particles of the body move in circles. e centres
of these circles lie on a straight line called the axis of
rotation, which is xed and perpendicular to the planes
of circle.
Torque and Angular Momentum
Torque is the turning eect of a force. If a force acting
on a object has a tendency to rotate the body about an
axis, the force is said to exert a torque on the body. It is
a vector quantity. In vector form,
Torque, τ=×
rF
In magnitude, W = r F sinq.
Here q is the angle between
rFand.
Torque has the same dimensions as that of work i.e.
[ML^2 T–2]. But work is a scalar quantity whereas torque
is a vector quantity.
By convention, anticlockwise moments are taken as
positive and clockwise moments are taken as negative.
Special Cases
(the rod will not rotate)
Ifqt= 0°, then = 0
O P
axis of rotation
r F
• (^) O P
If = 180°, then = 0
(the rod will not rotate)
qt
axis of rotationF
r
Ifq= 90°, thent=rF(maximum torque)
O P
axis of rotation F q= 90°
r
Note : Same force acting at the same point can produce
either anticlockwise or clockwise torque depending
upon the location of the axis of rotation as shown in
the gure.
A B
Axis of rotation
F
Anticlockwise torque
C
Axis of rotation
F
Clockwise torque
C
A B
Work Done by Torque
Work done, W = torque u angular displacement
= W × 'q
Power, P
dW
dt
d
dt
===τ
θ
τω
Angular momentum of a particle about a given axis is
the moment of linear momentum of the particle about
that axis. It is denoted by symbol
L.
Angular momentum
Lr=×p
In magnitude, L = rp sinq
where q is the angle between rpand.
Angular momentum is a vector quantity. Its SI unit is
kg m^2 s–1. Its dimensional formula is [ML^2 T–1].
Relationship between Torque and Angular
Momentum
Rate of change of angular momentum of a body is equal
to the external torque acting upon the body.
τext=
dL
dt